Fast time domain hyperbolic Radon transformsJuan I. Sabbione and Mauricio D. Sacchi

Department of Physics, University of Alberta

Summary

Seismic data can be modeled using a forward Radon transform. The coefficients needed to synthe-size the data in the Radon domain are estimated by solving an inverse problem. In this paper, wepresent a strategy to drastically speed up the time domain hyperbolic Radon transform. The methodis based on a restriction of the domain used to perform the inversion. Furthermore, this approachalso helps to focus the data representation and improves the signal-to-noise ratio enhancement. Weapply our algorithm to estimate the coefficients of the apex-shifted hyperbolic Radon transform. Ourfast implementation performs 20 times faster than the classical approach.

Introduction

Radon transforms have vast applications in seismic data processing (Trad et al., 2003). Originally,they were proposed to increase the SNR of seismic records (Thorson and Claerbout, 1985). Later,Radon transforms gained popularity as a technique for multiple attenuation (Hampson, 1986; Yilmaz,1989) and data reconstruction (Kabir and Verschuur, 1995). In recent years, multiple applicationshave been proposed. For instance, seismic data deblending (Ibrahim and Sacchi, 2014, 2015), anti-aliasing for reverse time migration (Wang and Nimsaila, 2014), de-ghosting (Wang et al., 2014) andmicroseismic signal detection and denoising (Sabbione et al., 2015). Two type of Radon transformshave been proposed. The classical implementation of the Radon transform in the frequency domainis often used to compute parabolic and linear Radon transforms. On the other hand, a time domainimplementation is generally reserved for hyperbolic and apex-shifted hyperbolic Radon transforms.In this paper, we will discuss a fast implementation of time domain hyperbolic Radon transforms andpay particular attention to the apex-shifted hyperbolic Radon transform.

Time domain Radon transforms are computed via iterative solvers. In each iteration of a typical solverto compute the Radon transform, one requires two linear operators in implicit form: the forward andthe adjoint Radon operators. This paper studies an on the flight strategy to compute both operatorsfaster. The proposed method uses the adjoint operator in conjunction with amplitude thresholding tospecify the set of coefficients in the Radon space that are active in the representation of the data.

Method

Common-shot gathers can be represented by a superposition of apex-shifted hyperbolas in the datadomain (t,x) according to t2 = τ2 +(x− a)2/v2. Here, τ is the two-way traveltime, and v and a areprocessing parameters used to focus reflections in the Radon domain. Following this assumption, wecan model the data with the time domain apex-shifted hyperbolic Radon transform:

d(t,x) = ∑v

∑a

m(τ =

√t2− (x−a)2

v2 ,v,a), (1)

mad j(τ,v,a) = ∑x

d(t =

√τ2 +

(x−a)2

v2 ,x). (2)

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Equation (1) represents the Radon forward operator and equation (2) the adjoint operator of thetransform. We can express them in matrix-times-vector form as follows

d = Lm (3)

mad j = LT d (4)

In our discussion, m(τ,v,a) represents the Radon space coefficients required to model the data viaexpression (1). Thus, the problem reduces to the estimation of m(τ,v,a) from d(t,x) (Thorson andClaerbout, 1985), and is solved by minimizing the following cost function

J(m) = ‖Lm−d‖22 +µR(m) . (5)

We have introduced a trade-off parameter µ and a regularization term R(m). The regularization termis needed to estimate a unique and stable solution. The classical solution of the Radon coefficientsvia damped-least squares adopts the l2 regularization term: (R(m) = ‖m|22). The high resolutionRadon transform, on the other hand, adopts either an l1 norm or a Cauchy (Sacchi and Ulrych,1995) constraint to impose sparsity on the distribution of Radon domain coefficients. Independentlyof the adopted regularization term, one must minimize equation (5) using an iterative solver wherethe operators L and LT must be provided in matrix-free form. In what follows, we define a strategy tospeed up the application of L and LT and, therefore, decrease the computational cost associated tothe estimation of m via iterative solvers.

Both the forward and adjoint operators can be computed by series of nested loops on the variables x,τ, v and a. If we maintain constant the size of the data d ∈ RNt×Nx , the cost of the application of L andLT is proportional to the size of Radon coefficients vector m ∈ RNτ×Nv×Na . In the proposed algorithm,we first determine which elements of the Radon domain are relevant in fitting the data. This is doneby applying thresholding to the Radon coefficients obtained via the adjoint operator: mad j = LT d.Clearly, according to equation (2), the coefficients mad j(τ,v,a) will be relatively large when integratingover reflections and small when integrating over random zero-mean noise. In the first pass of ouralgorithm we use amplitude thresholding of mad j(τ,v,a) to define a restricted Radon space A of active

coefficients. The latter is represented as A ={(τ,v,a) such that 1

Nx| mad j(τ,v,a) | > T

}, where T is

a threshold parameter that we need to set and satisfies 0 < T < 1. Note that the data has to benormalized to unity in order to adopt a bounded parameter T . The restricted operators LA and LT

Aare then used by our iterative solver to compute the Radon coefficients needed to model the data:

minv = argminm

[‖LAm−d‖2

2 +µ‖m‖22], (6)

d̂ = LAminv (7)

In our simulations, we have minimized the cost function given by (5) via the method of conjugategradients (Hestenes and Stiefel, 1952). It is worth noting that this strategy is similar to the algorithmproposed by Liu and Sacchi (2002). Nonetheless, unlike in our approach, Liu and Sacchi (2002)define areas of interest in both domains that change from iteration to iteration.

Example

To illustrate our method, we generated a synthetic data example from a simple 2D model with threereflectors. The data consists of 31 traces separated by ∆x= 50 m with sampling interval ∆t = 4 ms. Weused a Ricker wavelet with central frequency f0 = 20 Hz and added band-limited zero-mean randomnoise to the data with signal-to-noise ratio by amplitude equal to 2. For the first reflector, we setτ = 0.3 m/s, v = 1500 m/s and a = 390 m; for the second τ = 0.5 m/s, v = 2400 m/s and a = 460 m;

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Parameter Methodnv na HRT FHRT21 41 22.66 1.2821 61 33.59 1.6621 81 45.08 2.2531 41 33.18 1.6231 61 49.38 3.1431 81 67.31 3.2741 41 43.63 2.1741 61 66.10 3.4741 81 88.44 4.48

Table 1: Computing time re-quired to model the data withthe HRT and with the FHRT.Times are given in seconds.

1000

2000

3000

4000 −500

0

500

1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

a (m)v (m/s)

τ(s)

1Nx

| madj (τ , v, a) |

T

2T

Figure 1: Radon coefficients mad j(τ,v,a) thatform the restricted domain A. The colorbar scaleof the plot was limited to (T,2T ).

and τ = 0.8 m/s, v = 3000 m/s and a = −375 m for the third one. Regarding the inversion, the timeparameter of the Radon domain was scanned from τ = 0 to τ = 1.2 s, the velocity from v = 1000 tov = 4000 m/s and the apex from a =−500 to a = 1000 m.

We calculated the computational cost of the conventional hyperbolic Radon transform (HRT) andthe new fast hyperbolic Radon transform (FHRT). The results are shown in Table (1) in terms of thecomputing time needed to model the data for different sets of scanning parameters. In all cases, ∆τ

was set equal to ∆t such that all the time samples were scanned. Thus, we only varied the numberof velocities nv and the number of apexes na as shown in the table. The threshold parameter T wasset equal to 0.08, and we used the same trade-off parameter µ and truncated the inversion after 10iterations of the conjugate gradients inversion in all the examples. In general, we observe that theproposed method is about 20 times faster than the traditional one.

To gain an insight into the proposed method, Figure (1) shows the Radon coefficients that formthe subset A we used to invert the data. The colorbar scale was limited to (T,2T ) to facilitate thevisualization of the plot. This figure, and the following ones, correspond to the test with nv = 31and nx = 61. Note that only a low percentage of the Radon coefficients are used for the inversion.Figure (2) shows the three panels of the 3D Radon domain that correspond to the velocities used togenerate the data. Top panels show the Radon coefficients after equation (2) and the bottom ones theinversion using the restricted Radon domain A. Note that the plots that correspond to the inversionare highly focused on the reflection energy. Finally, the noisy input data and the output of the modeleddata are shown in Figure (3). The first plot corresponds to the original noisy data, the second plotto the data inverted using all the Radon coefficients (HRT) and the third one to the inversion usingthe restricted domain (FHRT). Note that new method not only speeds up the data processing, it alsonoticeably helps to focus the inversion around the reflection energies and improves the denoising.

Conclusions

The algorithm presented in this work considerably speeds up the application of the time domainRadon transform to reflection data. In addition, it helps to focus the inversion. This result is achievedby restricting the Radon coefficients used in the inversion. We identify the active set of coefficientsthat stack reflections using the adjoint operator followed by amplitude thresholding. This methodimproves the utilization of time-domain hyperbolic Radon transforms to perform multiple removal,denoising, interpolation and other applications.

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a)

a (m)

τ(s)

v1 = 1500 m/s

madj

−500 0 500 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

b)

a (m)

τ(s)

v2 = 2400 m/s

madj

−500 0 500 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

c)

a (m)

τ(s)

v3 = 3000 m/s

madj

−500 0 500 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

d)

a (m)

τ(s)

minv

−500 0 500 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

e)

a (m)

τ(s)

minv

−500 0 500 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

f)

a (m)

τ(s)

minv

−500 0 500 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 2: Panels of the 3D Radon domain for constant velocities v1 = 1500 m/s, v2 = 2400 m/s, andv3 = 3000 m/s. a), b) and c) Radon coefficients obtained using the adjoint operator (2). d), e) andf) Inverted model in the restricted domain.

−500 0 500 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

da)

Offset (m)

t(s)

−500 0 500 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

d̂HRTb)

Offset (m)

t(s)

−500 0 500 1000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

d̂FHRTc)

Offset (m)

t(s)

Figure 3: a) Original data with three reflections. b) Data predicted after traditional HRT using all theRadon coefficients. c) Data predicted after FHRT using restricted Radon domain.

Acknowledgements

The authors want to thank the sponsors of Signal Analysis and Imaging Group (SAIG) at the Univer-sity of Alberta.

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References

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